A one$-$year zero coupon bond issued by the government $($assumed to be risk$-$free$)$ with

face value $$100$ is traded for $$97\mathrm{.}5.$ Moreover, a $2-$year bond with face value $$100$ and $5\%$ annual coupons issued by the government is traded for $$101.$

$($a$)$ Determine the continuously compounded risk$-$free rates for $1$ year maturity and $2$ years maturity.

$($b$)$ We consider company Microsoft with $500$ shares that are traded for $$59\mathrm{.}04$

each. The company further has debts with notional $B=$$$80000$ in form of a zero coupon bond with maturity $T=2$ years. Assume the asset volatility is given by ${}_V=\sqrt[\phantom{2}]{0\mathrm{.}1},$ the drift is ${}_V=0\mathrm{.}05$ and that the current value of the zero coupon bond is $B_0=$$$70480.$ Compute both the actuarial $2-$year default probability and the risk$-$neutral $2-$year default probability of Microsoft in a Merton model.

$($c$)$ We consider a credit rating system with $4$ risk classes $($A$,$ B$,$ C and D$),$ where rating D represents the default. The one$-$year transition probability matrix is equal to:$P=([0\mathrm{.}94,0\mathrm{.}03,0\mathrm{.}02,0\mathrm{.}01],[0\mathrm{.}10,p,0\mathrm{.}07,1-p-0\mathrm{.}17],[0\mathrm{.}05,0\mathrm{.}15,0\mathrm{.}60,0\mathrm{.}20],[0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,1\mathrm{.}00])$

for pin$[0,0\mathrm{.}83].$ Assume the company Microsoft from $($a$)$ is initially $B-$rated. Determine the parameter $p$ such that the $2-$year probability of default coincides with the actuarial default probability in a Merton model that was computed in $($b$).$

$($d$)$ Consider company, Apple, and assume that a $2$ year zero coupon bond

$($face value $$100)$ of Apple is traded for price $$85.$ Assume that in case of default

during the first year payments are made already at the end of year $1,$ otherwise at the end of year $2.$ What is the constant recovery rate $R$ such that the market$-$implied risk$-$neutral $2-$year default probability for Apple coincides with the risk$-$neutral default probability of Microsoft that was computed in $($b$)?$